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For a polynomial F(t, A1, …, An) ∈ $$\mathbb{F}$$ p[t, A1, …, An] (p being a prime number) we study the factorization statistics of its specializations $$F\left({t,{a_1}, \ldots ,{a_n}} \right) \in {\mathbb{F}_p}\left[t \right]$$ with (a1, …, an) ∈ S, where $$S \subset \mathbb{F}_p^n$$ is a subset, in the limit p → ∞ and deg F fixed. We show that for a sufficiently large and regular subset $$S \subset \mathbb{F}_p^n$$ , e.g., a product of n intervals of length H1, …, Hn with $$\prod\nolimits_{i = 1}^n {{H_n} > {p^{n - 1/2 +\epsilon}}} $$ , the factorization statistics is the same as for unrestricted specializations (i.e., $$S = \mathbb{F}_p^n$$ ) up to a small error. This is a generalization of the well-known Polya-Vinogradov estimate of the number of quadratic residues modulo p in an interval.

Factorization statistics of restricted polynomial specializations over large finite fields